Free Interpolation Sets for Hölder Classes

Abstract

We carry out the simulations of pattern formation in a two-dimensional vibrated granular layer on an inclined base by molecular dynamics. It is found that the maximum amplitude of the pattern is greater at the lower part than at the higher part of the base, and is proportional to the thickness of the layer. Meanwhile, the wavelength varies non-monotonically as the inclined angle of the base is increased. SHI Xiao-Dong et al.Vol. 25

Boundary Interpolation Sets for Holomorphic Functions Smooth to the Boundary and BMO

“…This construction is the main step in the proof of the main theorem. This result is already present in Dynkin's paper [5], using some function theory on Lipschitz domains. Our contribution in this section is to prove these estimates just using the classical function theory in the unit disc.…”

“…But this is in fact equivalent to/ G As. Thus we can state (this is the theorem proved in [5] We will need the following lemma, which also gives the analogous result to (a) Gp{n\z) = OforO < n < p and z G E. Sufficiency. Suppose F is an ATW-set and <f> G CX(T).…”

“…A. Taylor and D. L. Williams proved that the B.I.-sets for A °° are exactly the ATW-sets. In [5], E. M. Dynkin has shown that the B.I.-sets for As are the A-sets. The Ap-case is, as far as we know, an open problem.…”

Sarason Conjecture on the Bergman Space

“…for all arcs I ⊂ T. If we assume the claim, then by Theorem 4 in [12] there exists an outer function w 1/2 in D such that…”

“…Finally, let E be the preimage of E 1 by the conformal map φ(z) = i 1+z 1−z , from the unit disc onto the upper half-plane. The following lemma is a direct application of a result in [12] and has been suggested to us by Konstantin Dyakonov.…”